Show unselected vertices in UV/Image editor?
Disable Sync Selection in the header toolbar of the UV/Image Editor.

Version 2.79: 



Version 2.8+:



This stops the UVs only being visible if their corresponding vertices are selected. Now, even if no vertices are selected in the 3D view the UV image editor will still show them. If you do then select some vertices in the 3D view, the equivalent UVs will be selected in the image editor.
Geometry hidden in the view-port will be hidden in the UV editor too.